Questions tagged [ideals]

An ideal is a subset of ring such that it is possible to make a quotient ring with respect to this subset. This is the most frequent use of the name ideal, but it is used in other areas of mathematics too: ideals in set theory and order theory (which are closely related), ideals in semigroups, ideals in Lie algebras.

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Show that $\frac{\mathbb{H}[\mathbb{Z}]}{ I_p}$ is isomorph in ${\mathbb{H}[\mathbb{Z}_p]}$

Consider ${I_p}$ = ${a_0 + a_1 i + a_2 j + a_3 k \in \mathbb{H}[\mathbb{Z}]}$, so that $p | a_i$
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Does every evaluation map $g: \Bbb{Z}[X,Y]/(\ker \pi) \to \Bbb{Z}$ factor some evaluation map $f:\Bbb{Z}[X,Y] \to \Bbb{Z}$?

Let $\Bbb{Z}[X,Y]$ be the polynomial ring. I know that every evaluation ring hom $f: \Bbb{Z}[X,Y] \to \Bbb{Z}$ is determined by where you send $X$ and $Y$. Let $I = (X^2 - Y^3)$ for example, but it ...
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Is the ideal $\langle x^2 + 1\rangle$ maximal in $\mathbb{Z}_3[x]$?

Is the ideal $\langle x^2 + 1\rangle$ maximal in $\mathbb{Z}_3[x]$? I am going about this by trying to prove that $\frac{\mathbb{Z}_3[x]}{\langle x^2 + 1 \rangle}$ is a field. I can prove commutative ...
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Showing $\langle x,y \rangle$ is a prime ideal of $K[x,y]$

The biggest thing that it stopping me from progressing in this question is the fact that I have two elements in the ideal, this topic it still very new to me. I can show that $I=\langle x \rangle$ and ...
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If $\operatorname{height}(P)=\operatorname{height}(J)$, then $P$ is minimal over $J$

Let $R$ be a regular local ring, and let $P$ be a prime ideal in $R$. Assume that $\operatorname{height}(P)=h$ and $P=(x_1,\ldots, x_k)$, for some $x_1,\ldots,x_k\in R$, and $k\geq h$. Let $J=(x_1,\...
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Show that $M$ is the set of all elements in $R$ which don't have a left multiplicative inverse.

Let $R$ be a ring with unity which contains exactly one maximal left ideal $M$. I'm trying to prove that $M$ is the set of all elements in $R$ which don't have a left multiplicative inverse. So I ...
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What is the educated way to find all maximal ideals of the ring?

Suppose I am given the ring, with basis elements $(e_1, e_2, \alpha, \beta)$. $e_1$ and $e_2$ are idemponents $e_1^2 = e_1, e_2^2= e_2$ and the following holds: $$ \alpha e_1 = \alpha = e_2 \alpha \...
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How do I find a finite list of polynomials $f_1(x,y),\dots, f_k(x,y)$ that generate the ideal $J \subset \mathbb R[x,y]$?

The Problem Consider the ring $R = \mathbb R[x,y]$ consisting of all polynomials in $x$ and $y$ with real coefficients, and let $J \subset R$ be the set of all polynomials $f(x,y) \in R$ such that $f(...
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Reduced Gröbner bases are minimal.

Let $F$ be a free module (of finite rank) over $S = k[x_1, \dots , x_r]$ with monomial order >. Let $M \subset F$ be a submodule and let $B = \{g_1, \dots , g_t\}$ be a Gröbner basis for $M.$ I ...
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Finding the syzygy (relation module) of a monomial ideal.

I have read pages 322-323 of "Commutative algebra, with a view toward algebraic geometry" by David Eisenbud, but it is still not much clear what are the steps of finding the syzygy. I am ...
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What exactly is a cokernel? What's the motivation behind that, and what's its use? And why is it a quotient module?

Having a homomorphism of $R$-modules $f\colon M \rightarrow N$, we say a cokernel is $N/(\operatorname{im} f)$ which is a $R$-module (quotient). Because $\operatorname{im} f\subset N$, this would mean ...
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Let $R$ be an arbitrary ring, $I\subset R$ an ideal. Why is $R/I$ an R-module?

We've defined that for any arbitrary ring $R$ and ideal $I\subset R$, $R/I$ is a module and is called a quotient R-module. But why is $R/I$ an R-module in the first place? I can't find anything on the ...
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Showing $x \in R$ is invertible if and only if $x$ is not contained in any maximal Ideal [duplicate]

Here $R$ is commutative. $\Rightarrow$ is fairly easy to prove as every Ideal containing a unit must be the entire Ring so it cannot be maximal. But I'm having some trouble with $\Leftarrow$, this is ...
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Prime Ideals in finite direct product of commutative Rings [duplicate]

Let $A_1, \dots , A_n$ be commutative unitary Rings and $A = \prod_{i=1}^{n} A_i$. Then every prime Ideal $\frak{p}$ $\subset A$ is of the form $\pi_i^{-1}(\frak{p}_i)$ where $\pi_i: A \to A_i$ are ...
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Let $ F $ be a field. Show that all polynomials with constant term $ a_0 = 0 $ form an ideal $ \langle x \rangle $ of $ F \langle x \rangle $.

Let $ F $ be a field. Show that all constant polynomials $ a_0 = 0 $ form an ideal $ \langle x \rangle $ of $ F \langle x \rangle $. My Try: Let $ F $ be a field. Suppose $ \langle x \rangle =\sum_{i=...
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An example of an ideal contained in union of ideals but not contained in any ideal seperatly [duplicate]

Find an example for a ring $R$ and ideals $I_1,...,I_n,J$ s.t $J\subseteq\bigcup_{i=1}^{n}I_i$ and $J\not\subseteq I_i$ for every $1\leq i\leq n$. I tried looking at some familiar rings like $\Bbb{Z},...
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Zariski topology closure of image of a homomorphism

Just started dealing with Zariski's topology on specta, and encountered the following question: $R,S$ commutative rings with unit and $\phi:R\rightarrow S$ homomorphism. Prove that the induced map $\...
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Determining whether $ \langle 2 - 5\omega_3\rangle$ is a prime ideal of $\mathbb{Z}[\omega_3]$ [closed]

$R = \mathbb{Z}[\omega_3]$ is defined to be $\{a + b\omega_3 : a,b \in \mathbb{Z}\}$ where $\omega_3$ is the cube root of unity. I want to see if $I = \langle 2 - 5\omega_3\rangle $ is a prime ideal. ...
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39 views

The null-ideal of a companion matrix

I am stuck with the following exercise (all required definitions and theorems are provided below): Let $R$ be a commutative ring, $f \in R[x]$ monic of degree $n$ and let $C_f \in M_n(R)$ be the ...
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Inclusion of ideal in a finite union of ideals [duplicate]

Say $R$ is a ring, and $I_1,...,I_n,J\subseteq R$ ideals, s.t $J\subseteq\bigcup_{i=1}^nI_i$. If there exists $\phi:K\rightarrow R$ homomorphism where $K$ is an infinite field, then there exists an $i$...
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Difference between subtractive and k- ideal of a semiring.

An ideal $I$ of a semiring $S$ is said to be k-ideal if $a\in I$ and $x\in S$, and if either $a+x\in I$ or $x+a\in I$ then $x\in I$. An ideal $I$ of a semiring $S$ is said to be $subtractive$ if $a\in ...
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Decomposition groups of non-prime ideals

In the Galois theory/algebraic number theory textbooks and lecture notes I've seen, the decomposition group and decomposition field are always defined for prime ideals. Is there a more general theory ...
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71 views

Proving that $[\mathcal F, \mathcal F]$ is an ideal.

Let $\mathcal{F}$ be a Lie algebra over a field $k.$ A lie subalgebra of $\mathcal{F}$ is a vector subspace $S \subset \mathcal{F}$ such that $[S,S] \subset S.$ An ideal of $\mathcal{F}$ is a vector ...
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26 views

Example of strict inclusion with respect to the product of initial ideals

I solved the following problem: Show that for any monomial order $>$ it is true that $\mathrm{in}_{>}(I)\mathrm{in}_{>}(J)\subseteq \mathrm{in}_{>}(IJ)$ for any two ideals $I,J$ of $k[x_{1}...
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53 views

Surjectivity of a given map

Let $R$ be a local ring. Then $f:R^\times\to (R/I)^\times$ induced by quotient map $R\to R/I$ is surjective. (Here, $I\subsetneq R$ is an ideal.) Let $a+I\in (R/I)^\times$. Then there is $b+I\in (R/I)...
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$R$ is finitely generated?

I have seen many books using the idea that a commutative semi-simple ring with unity is finitely generated as an $R-$module but I do not understand why this is correct. Any elaboration will be ...
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Spectra of cartesian product of rings isomorphic to disjoint union of spectra

I want to prove that $\operatorname{Spec}(R_1\times R_2)\cong\operatorname{Spec}(R_1)\sqcup\operatorname{Spec}(R_2).$ I tried proving that every prime ideal in $R_1\times R_2$ is of the form $p\times ...
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Show that $\langle 2,1+\sqrt{-5}\rangle \langle3,1+\sqrt{-5} \rangle = \langle1-\sqrt{-5} \rangle$

In $\mathbb Z[\sqrt{-5}]$ I compute $\langle 2,1+\sqrt{-5} \rangle \langle 3,1+\sqrt{-5} \rangle = \langle 6,2+2\sqrt{-5},3+3\sqrt{-5},-4+2\sqrt{-5} \rangle $ So there must be $a+b\sqrt{-5}\in\mathbb ...
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1answer
120 views

Maximal ideal with given condition

I am reading the proof by B. Banaschewski of "Krull implies Zorn." I am having difficulties filling in the details in one of the steps. We start with a partition $\mathfrak U$ of an ...
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If I,J radicals then I+J radical? [closed]

Assume I, J are radicals in commutative ring is it true that $I+J$ radical?
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75 views

Relations among roots of polynomials

Suppose we have a polynomial $p \in \mathbb{Q}[x]$ and we consider the roots $\alpha_0,\dots,\alpha_d \in \mathbb{C}$ of $p$. Then, $p$ imposes some polynomial relations on the $\alpha_i$. Let us ...
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Understanding this definition of a relation on an ideal

In Kunen's Set Theory, he gives a definition of $R_{\mathcal{I}}$ where $\mathcal{I}$ is an ideal. If $R$ is a relation on $B$ and $\mathcal{I}$ is an ideal with dual filter $\mathcal{F}$ on $A$ and $...
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Ring of Formal Laurent Series which are Dedekind domains

Let $R$ be an integral domain and $R((x))$ be the ring of formal Laurent series over $R$. (The answer to this question has a good explanation for our ring.) Is it true that $R$ is a Dedekind domain ...
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How many elements are there in $\mathbb{Z}[i]/I$

I am studying Ideals and Factor Rings and seem to get stuck on a specific type of questions. What is the general technique to attack questions like : How many elements are there in $\mathbb{Z}[i]/I$ ...
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21 views

Are there specific properties defining the kernel of a quotient ring?

I'm asking because the correction of an exercise of a course I'm following (commutative algebra) has the following conclusion: Let I be an ideal of a commutative ring A, and $I[X]=\{\sum_{k=0}^{n}i_{...
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How to prove if $\mathfrak{a}$ is a radical ideal then $(\mathfrak{a}:\mathfrak{b})$ is also a radical ideal. [closed]

I am given $\mathfrak{a}$ and $\mathfrak{b}$ are ideals with $\mathfrak{a}$ a radical ideal, i.e., $\mathfrak{a} = \sqrt{\mathfrak{a}}$. How to prove the quotient $(\mathfrak{a}{\,:\,}\mathfrak{b})$ ...
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Radical ideal of not necessarily Noetherian ring.

Let $(A,\mathfrak{m})$ be a local integral domain (not necessarily Noetherian) with Krull dimension 1. Pick an element $\eta \in \mathfrak{m}$. How can I show that $I=(\eta)$ is a primary ideal? In ...
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28 views

Examples of ideals of $K$- algebras that do not have a basis?

I wanted to be more familiar with the idea of $K$-algebra etc and so I wanted to collect some examples for an ideal that do not have a basis. Definition: A $K$-algebra is a ring who is also a $K$-...
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Why do the k-fold products of the germs of functions that vanish on a neighborhood of a point m form a descending sequence of ideals?

I'm going through Warner's Foundations of Differentiable Manifolds and Lie Groups, and he loves the idea of defining the tangent vectors at $p \in M$ using the germs of functions that agree on a ...
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Ideals versus normal subgroups

Let $\phi : R \to R'$ be a ring homomorphism with $S$ a subring of $R$. Let $I, J$ be ideals of $R$ with $I \subseteq J \subseteq R$. There is a one-to-one correspondence between the ideals of $R$ ...
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30 views

Left Ideals in $C^*$ algebra

Let $A$ be $C^*$-algebra, $L$ be a maximal left ideal, $\Lambda(A)= \{ L \subset A\}$, and $rad(A) = \cap_{L \in \Lambda(A)} L$. I'm trying to understand this Lemma. But I'm stuck on understanding ...
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23 views

Are maximal ideals and principal ideals independent?

How can I show that the properties maximal and principal can be true independently for an ideal? I can find one example of each: A maximal principal ideal A maximal non-principal ideal A non-maximal ...
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29 views

Relationship between membership in a polynomial ideal, and derivable statements about solutions to the system of equations

Consider a set of polynomials $P$ in the polynomial ring of $n$ variables $\mathbb{C}[x_1,...,x_n]$, and let $I$ be the ideal generated by the polynomials in $P$. If for any zero (a solution) $x_1,...,...
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36 views

Questions about definition of principal ideals

The definition for an ideal to be principal is: Let $R$ be a commutative ring with a unit element. An ideal $I$ of $R$ is $\textbf{principal}$ if there exist $a\in R$ such that $I=\{ar\mid r\in R\}$. ...
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86 views

Semisimple Rings are Noetherian

I have seen the statement that if $R$ is a commutative ring and $R$ is a semisimple ring, then $R$ is Noetherian several times while reading through some algebra resources; however, I have never seen ...
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21 views

Distribution of prime/irreducible ideals

There are results on the distribution of prime ideals in any arbitrary ring of integers inside a number field? Are there any similar results for irreducible ideals in the ring of integers? Also, do we ...
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40 views

I don't understand the definition of $D\left(f\right)=\left\{p\:\in \:Spec\:R\::\:f\:\notin \:p\right\}=Spec\:R\:\setminus \:V\left(f\right)$

Where $V\left(f\right)=\left\{a\in \mathbb{K}^n\::\:f\left(a\right)=0\right\}$ and $f\in \:\mathbb{K}\left[X_1,\:...,\:X_n\right]$ I don't understand why $$\left\{p\:\in \:Spec\:R\::\:f\:\notin \:p\...
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31 views

What was the motivation and what is the geometric interpretation of Zariski Topology and Prime Spectrum?

Those are the definitions given to us during our lecture: Zariski Topology We call a Zariski Topology in $\mathbb{K}^n$ a family of complements of $V\left(I\right)=\left\{a\in \mathbb{K}^n\::\:\...
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2answers
37 views

Prime ideals of the ring $\Bbb C[x,y]$, proof given in Vakil's book.

I am reading Vakil's Algebraic Geometry lecture note, and the following is the proof of the statement that any non-principal prime ideal of $\Bbb C[x,y]$ is of the form $(x-a,y-b)$ for some $a,b\in \...
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1answer
29 views

Understanding the Generalised Chinese Remainder Theorem

The generalised version of the theorem I'm working with is: If $R$ is a commutative Ring with unity and $I_1,\ldots , I_n$ ideals of $R$, then the map $$\phi: R \to \frac{R}{I_1}\times\ldots\times \...

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